Binary Liquid–Liquid Equilibrium

Introduction

Two liquids do not always mix into one phase at every composition. At a particular temperature, an intermediate overall composition may separate into an A-rich liquid and a B-rich liquid. A liquid–liquid phase diagram shows the compositions of these coexisting phases, the overall compositions that separate, and sometimes a critical solution temperature where the distinction disappears.

Core explanation

At fixed temperature and pressure, a binary mixture can minimise its Gibbs energy by forming two liquid phases α and β with different compositions. Equilibrium requires μ A^α = μ A^β and μ B^α = μ B^β. On a molar Gibbs-energy-versus-composition plot, a common tangent touching the curve at x A^α and x A^β identifies the coexisting compositions. A straight line joining the two tangent points can give a lower total G for an intermediate overall composition than the homogeneous curve itself.

The binodal marks the coexistence compositions as temperature changes. At a temperature inside a miscibility gap, a horizontal tie line connects the A-rich and B-rich endpoints. An overall composition between them separates into two phases; its location along the tie line determines phase amounts by material balance. Outside the gap, one homogeneous liquid is stable. At a critical solution or consolute point, the two coexisting compositions merge and the tie line shrinks to zero length.

Some mixtures have an upper critical solution temperature: above it they are fully miscible over the relevant compositions, while below it a gap exists. Other systems can have a lower critical solution temperature or more complex closed-loop behaviour. These are empirical possibilities, not universal rules derivable from merely knowing that two liquids have different polarities.

The binodal should be distinguished from the spinodal. Between binodal and spinodal, a homogeneous state may be metastable; small composition fluctuations raise G, but a sufficiently large fluctuation can nucleate separation. Inside the spinodal, the homogeneous free-energy curvature is negative in a simple composition model, so infinitesimal fluctuations lower G and separation can begin without a nucleation barrier. Real kinetics, interfaces and finite size influence how quickly and visibly these processes occur.

Step-by-step reasoning

Fix T and p, locate the overall composition and compare it with binodal endpoints. If inside the two-phase region, draw a tie line and read each liquid's composition. Use overall material balance for amounts. If asked about local stability, inspect free-energy curvature or a labelled spinodal rather than assuming the binodal alone tells whether infinitesimal fluctuations grow.

Visual explanation

Draw a temperature–composition diagram with a dome-shaped miscibility gap ending at an upper critical solution point. Put an A-rich liquid on the left and a B-rich liquid on the right, joined by a horizontal tie line. Inside the dome sketch a smaller spinodal curve. Beside it show a double-well Gibbs-energy curve and a common tangent between its two stable minima.

Real-world analogy

Some mixed crowds spontaneously organise into two groups with different proportions of members rather than staying evenly distributed. A tie line records each group's composition, while the overall crowd proportion determines group sizes. The analogy is descriptive; liquid separation follows Gibbs-energy competition between mixing entropy and molecular interactions.

Real-world example

Certain pairs of organic liquids are only partly miscible over a range of temperatures. They form two liquid layers with different compositions, not necessarily pure A on one side and pure B on the other. Extraction processes exploit those distinct phase compositions to move a solute preferentially between layers.

Why?

Mixing entropy generally favours one homogeneous phase, while unfavourable unlike interactions can favour segregation. If the resulting Gibbs-energy curve is nonconvex, a weighted combination of two compositions can lie below the homogeneous free energy at an intermediate composition. Common-tangent construction and equal component chemical potentials express the same equilibrium condition.

Common misconception

Two separated liquid phases are rarely pure individual components; each can dissolve some of the other. The binodal is not the same as the spinodal, and crossing the binodal need not cause immediate visible separation if nucleation is slow. A critical solution temperature can be upper or lower depending on the system.

Worked example

At one T,p, suppose tie-line liquid endpoints have x A^α = 0.20 and x A^β = 0.80. A sample has overall z A = 0.35. Let f β be its β-phase mole fraction; material balance gives 0.35 = (1−f β)0.20 + f β0.80 = 0.20 + 0.60f β. Hence f β = 0.25 and f α = 0.75. The two phase compositions remain 0.20 and 0.80; they do not each equal the overall 0.35.

Quick check

1. What happens to a liquid–liquid tie line at its critical solution point? Answer: Its two endpoint compositions merge, so the tie-line length tends to zero and the two liquids become indistinguishable as separate phases at that point.

Exam focus

Label liquid phase compositions and overall composition separately. Use equal μ for each component or a common tangent to justify coexistence. Apply the lever rule to amounts only after endpoints are known. Distinguish metastable binodal-to-spinodal region from locally unstable spinodal interior if the diagram or free-energy curve provides that information.

Advanced insight

The curvature ∂²g/∂x² of a homogeneous molar free-energy model is a local stability test. At a spinodal it reaches zero; inside, negative curvature means infinitesimal composition fluctuations can reduce free energy. Interfacial energy and diffusion kinetics determine the actual separation pattern and timescale.

Summary

A binary liquid miscibility gap contains overall compositions that separate into two liquids. Binodal endpoints and tie lines give their compositions; the lever rule gives amounts. Critical solution points merge the phases. The spinodal marks local instability and differs from the coexistence binodal, reflecting nucleation versus fluctuation-driven separation.

Practice questions

1. A tie line connects x B = 0.10 and 0.70. If overall z B = 0.40, what fraction lies in the B-rich phase? Answer: f = (0.40 − 0.10)/(0.70 − 0.10) = 0.50 by material balance, assuming mole-fraction phase amounts. 2. Can a homogeneous liquid between binodal and spinodal persist for a while? Answer: Yes. It can be metastable: small fluctuations are resisted, but a sufficiently large nucleus of the separated phase lowers total G. 3. Does an upper critical solution temperature mean the liquids separate at every temperature above it? Answer: No. In the usual upper-critical case, the miscibility gap closes at the critical temperature and the liquids are fully miscible above it over the relevant compositions.